Distributive Equations of Implications Based on Continuous Triangular Norms (I)

Authors:
Feng Qin;Michał Baczynski;Aifang Xie
Affiliations:
College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nanchang, China;Institute of Mathematics, University of Silesia, Poland;School of Mathematics, Shandong University, 250100 Jinan, China
Venue:
IEEE Transactions on Fuzzy Systems
Year:
2012

Citing 0
Cited 4

On distributive equations of implications and contrapositive symmetry equations of implications based on a continuous t-Norm

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On two distributivity equations for fuzzy implications and continuous, Archimedean t-norms and t-conorms

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Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms

Information Sciences: an International Journal
Distributivity equations of implications based on continuous triangular conorms (II)

Fuzzy Sets and Systems

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Abstract

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this paper, we explore the distributive equations of implications. In detail, by means of the sections of $I$ , we give out the sufficient and necessary conditions of solutions for the distributive equation of implication $I(x,T_1(y,z))=T_2(I(x,y),I(x,z))$, when $T_1$ is a continuous but not Archimedean triangular norm, $T_2$ is a continuous and Archimedean triangular norm, and $I$ is an unknown function. This obtained characterizations indicate that there are no continuous solutions for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that $I$ is continuous except for the point (0,0), we get its complete characterizations. Here, it should be pointed out that these results make differences with recent results that are obtained by Baczyński and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.